Controlled surgery and $\mathbb{L}$-homology

Abstract

This paper presents an alternative approach to controlled surgery obstructions. The obstruction for a degree one normal map $(f,b):M^n\to X^n$ with control map $q:X^n\to B$ to complete controlled surgery is an element ${\sigma }^c(f,b)\in H_n(B,\mathbb{L})$, where $M^n,X^n$ are topological manifolds of dimension $n\ge 5$. Our proof uses essentially the geometrically defined $\mathbb{L}$-spectrum as described by Nicas (going back to Quinn) and some well known homotopy theory. We also outline the construction of the algebraically defined obstruction, and we explicitly describe the assembly map $H_n(B,\mathbb{L})\to L_n({\pi }_1(B))$ in terms of forms in the case $n\equiv 0(4)$. Finally, we explicitly determine the canonical map $H_n(B,\mathbb{L})\to H_n(B,L_0)$.